Example12.4.2
Simplify the expression \((1-7i)+(5+4i)\text{.}\)
\begin{align*}
(1-7i)+(5+4i)\amp=1+5-7\highlight{i}+4\highlight{i}\\
\amp=6-3\highlight{i}
\end{align*}
Section12.4Complex Number Operations
¶Complex numbers 1 en.wikipedia.org/wiki/Complex_number#Applications are used in many math, science and engineering applications. In this section, we will learn the basics of complex number operations.
Subsection12.4.1Adding and Subtracting Complex Numbers
Adding and subtracting complex numbers is just like combining like terms. We combine the terms that are real and the terms that are imaginary. Here are some examples
Example12.4.3
Simplify the expression \((3-10i)-(4-6i)\text{.}\)
\begin{align*}
(3-10i)-(4-6i)\amp=3-10\highlight{i}-4+6\highlight{i}\\
\amp=-1-4\highlight{i}
\end{align*}
Example12.4.4
Simplify the expression \((8+2i)-(5+3i)\text{.}\)
\begin{align*}
(8+2i)-(5+3i)\amp=8+2\highlight{i}-5-3\highlight{i}\\
\amp=3-\highlight{i}
\end{align*}
Subsection12.4.2Multiplying Complex Numbers
Now let's learn how to multiply complex numbers. It is very similar to multiplying polynomials.
Example12.4.5
Simplify the expression \(2i(3-2i)\text{.}\)
We distribute the \(2i\) to both terms, then we simplify any powers of \(i\text{.}\)
\begin{align*}
2i(3-2i)\amp=2i\cdot3-2i\cdot2i\\
\amp=6i-4i^2\\
\amp=6i-4(-1)\\
\amp=6i+4\\
\amp=4+6i
\end{align*}
Note that we always write a complex number in standard form, which is \(a+bi\text{.}\)
When we multiply two complex numbers we can use the distributive method, FOIL method, or generic rectangles. Here is an example of each method.
Example12.4.6
Multiply \((1+5i)(2-7i)\text{.}\)
We will use the distributive method to multiply the two binomials.
\begin{align*}
(1+5i)(2-7i)\amp=2(1+5i)-7i(1+5i)\\
\amp=2+10i-7i-35i^2\\
\amp=2+10i-7i-35(-1)\\
\amp=2+3i+35\\
\amp=37+3i
\end{align*}
Example12.4.7
Expand and simplify the expression \((3-4i)^2\text{.}\)
Solution
Figure12.4.8Using the FOIL method to expand \((3-4i)^2\)
We will use the FOIL method to expand this perfect square.
\begin{align*}
(3-4i)^2\amp=(3-4i)(3-4i)\\
\amp=9-12i-12i+16i^2\\
\amp=9-24i+16(-1)\\
\amp=9-24i-16\\
\amp=-7-24i
\end{align*}
Example12.4.9
Multiply \((3+4i)(3-4i)\text{.}\)
Solution
Figure12.4.10Using Generic Rectangle Method to multiply \((3+4i)(3-4i)\)
We will use the Generic Rectangle Method to multiply those two binomials.
\begin{align*}
(3+4i)(3-4i)\amp=9+12i-12i-16i^2\\
\amp=9-16(-1)\\
\amp=9+16\\
\amp=25
\end{align*}
As the last example shows, it is possible to multiply two complex numbers and get a real number result. Notice that the middle terms, \(12i\) and \(-12i\text{,}\) are opposites, which makes the result a real number. This happens when we multiply a sum and difference of the same real and imaginary parts, called complex conjugates. This pair of factors results in the difference of squares:
\begin{equation*}
(a+b)(a-b)=a^2-b^2\text{.}
\end{equation*}
Example12.4.11
Here is an example using the sum and difference formula to multiply \((5+2i)(5-2i)\text{:}\)
\begin{align*}
(5+2i)(5-2i)\amp=5^2-(2i)^2\\
\amp=25-4i^2\\
\amp=25-4(-1)\\
\amp=25+4\\
\amp=29
\end{align*}
Example12.4.12
Multiply \((7-9i)(7+9i)\text{.}\)
\begin{align*}
(7-9i)(7+9i)\amp=(7)^2-(9i)^2\\
\amp=49-81i^2\\
\amp=49-81(-1)\\
\amp=49+81\\
\amp=130
\end{align*}
Subsection12.4.3Dividing Complex Numbers
When we divide by \(i\) we use a process that is similar to rationalizing the denominator. We use the property \(\sqrt{x}\cdot\sqrt{x}=x\) when we rationalize the denominator, and we use the property \(i\cdot i=-1\) when we have complex numbers. Let's compare these two problems \(\frac{2}{\sqrt{2}}\) and \(\frac{2}{i}\text{:}\)
\begin{align*}
\frac{2}{\sqrt{2}}\amp=\frac{2\multiplyright{\sqrt{2}}}{\sqrt{2}\multiplyright{\sqrt{2}}}\\
\amp=\frac{2\sqrt{2}}{2}\\
\amp=\sqrt{2}
\end{align*}
\begin{align*}
\frac{2}{i}\amp=\frac{2\multiplyright{i}}{i\multiplyright{i}}\\
\amp=\frac{2i}{-1}\\
\amp=-2i
\end{align*}
Example12.4.13
Rationalize the denominator in the expression \(-\frac{7}{4i}\text{.}\)
\begin{align*}
-\frac{7}{4i}\amp=-\frac{7\multiplyright{i}}{4i\multiplyright{i}}\\
\amp=-\frac{7i}{4(-1)}\\
\amp=\frac{7i}{4}\\
\amp=\frac{7}{4}i
\end{align*}
Example12.4.14
Rationalize the denominator in the expression \(\frac{5}{3i}\text{.}\)
\begin{align*}
\frac{5}{3i}\amp=\frac{5\multiplyright{i}}{3i\multiplyright{i}}\\
\amp=\frac{5i}{3(-1)}\\
\amp=-\frac{5i}{3}\\
\amp=-\frac{5}{3}i
\end{align*}
When the denominator is in the form \(a+bi\text{,}\) we need to use the complex conjugate to remove the imaginary terms from the denominator. Here is an example.
Example12.4.15
Simplify the expression \(\frac{1}{4+3i}\text{.}\)
Solution
To get a real result in the denominator we multiply the numerator and denominator by \(4-3i\text{,}\) and we have:
\begin{align*}
\frac{1}{4+3i}\amp=\frac{1}{4+3i}\multiplyright{\frac{(4-3i)}{(4-3i)}}\\
\amp=\frac{4-3i}{16-12i+12i-9i^2}\\
\amp=\frac{4-3i}{16-9(-1)}\\
\amp=\frac{4-3i}{16+9}\\
\amp=\frac{4-3i}{25}\\
\amp=\frac{4}{25}-\frac{3}{25}i
\end{align*}
Note that we always write complex numbers in standard form which is \(a+bi\text{.}\)
Now we can divide two complex numbers as in the next example.
Example12.4.16
Simplify the expression \(\frac{1+2i}{2-4i}\text{.}\)
Solution
To divide complex numbers, we rationalize the denominator using the conjugate \(2+4i\text{:}\)
\begin{align*}
\frac{1+2i}{2-4i}\amp=\frac{(1+2i)}{(2-4i)}\multiplyright{\frac{(2+4i)}{(2+4i)}}\\
\amp=\frac{2+4i+4i+8i^2}{4+8i-8i-16i^2}\\
\amp=\frac{2+8i+8(-1)}{4-16(-1)}\\
\amp=\frac{2+8i-8}{4+16}\\
\amp=\frac{-6+8i}{20}\\
\amp=\frac{-6}{20}+\frac{8i}{20}\\
\amp=-\frac{3}{10}+\frac{2}{5}i
\end{align*}
Example12.4.17
Simplify the expression \(\frac{4-7i}{5+i}\text{.}\)
Solution
To divide, we rationalize the denominator using the conjugate \(5-i\text{:}\)
\begin{align*}
\frac{4-7i}{5+i}\amp=\frac{(4-7i)}{(5+i)}\multiplyright{\frac{(5-i)}{(5-i)}}\\
\amp=\frac{20-4i-35i+7i^2}{25-5i+5i-i^2}\\
\amp=\frac{20-39i+7(-1)}{25-1(-1)}\\
\amp=\frac{20-39i-7}{25+1}\\
\amp=\frac{13-39i}{26}\\
\amp=\frac{13}{26}-\frac{39i}{26}\\
\amp=\frac{1}{2}-\frac{3}{2}i
\end{align*}
SubsectionExercises
Adding and Subtracting Complex Numbers
1
Add up the following complex numbers:
\(\displaystyle{ ({3-6i})+({-10+3i}) = }\)
2
Add up the following complex numbers:
\(\displaystyle{ ({5+11i})+({-4+5i}) = }\)
3
Subtract the following complex numbers:
\(\displaystyle{ ({8+2i})-({11-11i}) = }\)
4
Subtract the following complex numbers:
\(\displaystyle{ ({11-7i})-({-4+8i}) = }\)
5
Write the complex number in standard form.
\begin{equation*}
(-10+8i)+(6+i)
\end{equation*}
6
Write the complex number in standard form.
\begin{equation*}
(-7+i)+(-7-5i)
\end{equation*}
7
Write the complex number in standard form.
\begin{equation*}
(-5-7i)+(2+10i)
\end{equation*}
8
Write the complex number in standard form.
\begin{equation*}
(-3+7i)+(-10+4i)
\end{equation*}
9
Write the complex number in standard form.
\begin{equation*}
(0)-(-1-2i)
\end{equation*}
10
Write the complex number in standard form.
\begin{equation*}
(2-7i)-(7-8i)
\end{equation*}
11
Write the complex number in standard form.
\begin{equation*}
(4+6i)-(-5+8i)
\end{equation*}
12
Write the complex number in standard form.
\begin{equation*}
(7-i)-(4+2i)
\end{equation*}
Multiplying Complex Numbers
13
Multiply the following complex numbers:
\(\displaystyle{ {i}({11-10i}) = }\)
14
Multiply the following complex numbers:
\(\displaystyle{ {i}({-11+6i}) = }\)
15
Multiply the following complex numbers:
\(\displaystyle{ ({-9-2i})({11+6i}) = }\)
16
Multiply the following complex numbers:
\(\displaystyle{ ({-6-11i})({-4-i}) = }\)
17
Multiply the following complex numbers:
\(\displaystyle{ ({-3+5i})^{2} = }\)
18
Multiply the following complex numbers:
\(\displaystyle{ ({-3-8i})^{2} = }\)
19
Multiply the following complex numbers:
\(\displaystyle{ ({2-12i})({2+12i}) = }\)
20
Multiply the following complex numbers:
\(\displaystyle{ ({5+4i})({5-4i}) = }\)
21
Write the complex number in standard form.
\begin{equation*}
(7-4i)(-1-9i)
\end{equation*}
22
Write the complex number in standard form.
\begin{equation*}
(9+10i)(7+7i)
\end{equation*}
23
Write the complex number in standard form.
\begin{equation*}
(-10+3i)(-5+i)
\end{equation*}
24
Write the complex number in standard form.
\begin{equation*}
(-7-5i)(4-5i)
\end{equation*}
Dividing Complex Numbers
25
Rewrite the following expression into the form of a+b\(i\text{:}\)
\(\displaystyle{ \frac{-12}{i} = }\)
26
Rewrite the following expression into the form of a+b\(i\text{:}\)
\(\displaystyle{ \frac{-8}{i} = }\)
27
Rewrite the following expression into the form of a+b\(i\text{:}\)
\(\displaystyle{ \frac{{9-8i}}{{-1+2i}} = }\)
28
Rewrite the following expression into the form of a+b\(i\text{:}\)
\(\displaystyle{ \frac{{-11-3i}}{{-1+2i}} = }\)
29
Rewrite the following expression into the form of a+b\(i\text{:}\)
\(\displaystyle{ \frac{{3+i}}{{5+6i}} = }\)
30
Rewrite the following expression into the form of a+b\(i\text{:}\)
\(\displaystyle{ \frac{{5-5i}}{{-5+2i}} = }\)
31
Write the complex number in standard form.
\begin{equation*}
\frac{9+7i}{2-4i}
\end{equation*}
32
Write the complex number in standard form.
\begin{equation*}
\frac{-10}{-10-10i}
\end{equation*}
33
Write the complex number in standard form.
\begin{equation*}
\frac{-7-7i}{-1+6i}
\end{equation*}
34
Write the complex number in standard form.
\begin{equation*}
\frac{-5+6i}{8-4i}
\end{equation*}